On the Volume of Unions of Translates of a Convex Set

Rehder, Wulf

doi:10.2307/2321212

Cited by 3 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Clearly, if K is an unconditional convex body in E d , then K is symmetric about the origin o of E d . Our third main result is a generalization of the Kneser-Poulsen-type results published in [Bou28] and [Reh80].…”

Section: Introductionsupporting

confidence: 64%

The Kneser–Poulsen Conjecture for Special Contractions

Bezdek

Naszódi

2018

Discrete Comput Geom

View full text Add to dashboard Cite

The Kneser-Poulsen Conjecture states that if the centers of a family of N unit balls in E d is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions.First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that N ≥ (1 + √ 2) d . Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union.Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.2010 Mathematics Subject Classification. 52A20,52A22.

show abstract

Section: Introductionsupporting

confidence: 64%

The Kneser–Poulsen Conjecture for Special Contractions

Bezdek

Naszódi

2018

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…For a different approach to the case when the configurations are similar see [6] by Bouligand. When the sets forming the union are not spherical balls, but translates of a convex set, then in [22], Rehder shows that the volume of the union does not decrease when the sets are dilated. However, a general expansive rearrangement of convex sets different from ellipsoids can have the volume of the intersection (union) increase (decrease), as shown in [19] by Meyer, Reisner and Schmuckenschläger.…”

Section: Resultsmentioning

confidence: 99%

Pushing disks apart - the Kneser-Poulsen conjecture in the plane

Bezdek¹,

Connelly²

2002

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

show abstract

“…For the asymptotic behavior of such integrals it will prove important to know the infinitesimal growth of these molecules, i.e. we need lemmas for the function It is intuitively clear that g(z, r) is monotonic in t. This has been proved on p. 324 of [2] for a sphere, and, by a straightforward argument, in [7] for any convex set K. In addition to this, we need the growth-rate of g(z, r) near t = 0, i.e. the value of…”

Section: Unions Of Spheres and Properties Of Integralsmentioning

confidence: 99%